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I shall describe some joint work with Vladimir Remeslennikov and Ilia
Kazachkov.
Partially commutative groups are groups given by a presentation determined by
a graph:
vertices are generators and edges define commutation relations.
Divisbility and orthogonal systems are tools developed to study these groups.
Using them we have descriptions of centralisers of subsets, a good understanding
of the centraliser lattice in terms of the underlying graph and have made good
progress towards classifying the universal theory of these groups as well as
their automorphism groups.
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I will discuss Garside's representation of elements of the braid group in
terms of &quot;half- twists&quot; and the corresponding solution to the Conjugacy Problem,
originally posed by Artin. If time permits, I will discuss some geometric
implications of this result.
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I will discuss the proof that the exponential algebraic closure operator on
the
complex exponential field is isomorphic to the pregeometry which controls the
&quot;pseudoexponential&quot; field.
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